{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:JGVDTCD5WQ6AKHYBXOTH436IF6","short_pith_number":"pith:JGVDTCD5","canonical_record":{"source":{"id":"1701.04347","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-01-16T16:21:30Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"7729570c7d14161d1fb6699c5265ab15870458ccd56f7365bdc7920182c0042b","abstract_canon_sha256":"040766acb811a7ce444f1b6d1e48198f8a488c5a33b77b46b288b00adbd06235"},"schema_version":"1.0"},"canonical_sha256":"49aa39887db43c051f01bba67e6fc82fad561a20838c9436370850350bcf48e0","source":{"kind":"arxiv","id":"1701.04347","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.04347","created_at":"2026-05-18T00:11:11Z"},{"alias_kind":"arxiv_version","alias_value":"1701.04347v4","created_at":"2026-05-18T00:11:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04347","created_at":"2026-05-18T00:11:11Z"},{"alias_kind":"pith_short_12","alias_value":"JGVDTCD5WQ6A","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"JGVDTCD5WQ6AKHYB","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"JGVDTCD5","created_at":"2026-05-18T12:31:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:JGVDTCD5WQ6AKHYBXOTH436IF6","target":"record","payload":{"canonical_record":{"source":{"id":"1701.04347","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-01-16T16:21:30Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"7729570c7d14161d1fb6699c5265ab15870458ccd56f7365bdc7920182c0042b","abstract_canon_sha256":"040766acb811a7ce444f1b6d1e48198f8a488c5a33b77b46b288b00adbd06235"},"schema_version":"1.0"},"canonical_sha256":"49aa39887db43c051f01bba67e6fc82fad561a20838c9436370850350bcf48e0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:11.799492Z","signature_b64":"sSOPrOFfFA5eQ1yWMiwyIu7JFxwyoQbZUoK91H4GOf1xLgxT0BrgXlNVExhqFv+Z74NhwAPBP7GJlijoBgv5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49aa39887db43c051f01bba67e6fc82fad561a20838c9436370850350bcf48e0","last_reissued_at":"2026-05-18T00:11:11.798596Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:11.798596Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1701.04347","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:11:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wpvJsrvFbK7HYUWUL5lqrUo6BnjSYn3OU7C2e0js4OK8Z/k8R9FwAiVkhcqnHyVYcsVoIz24hgqmitvG3QC8DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T12:19:46.481206Z"},"content_sha256":"22b5173426fc5c47b8a890c76e2f3094f2684b2f8b094f3629ae0967ef386eb1","schema_version":"1.0","event_id":"sha256:22b5173426fc5c47b8a890c76e2f3094f2684b2f8b094f3629ae0967ef386eb1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:JGVDTCD5WQ6AKHYBXOTH436IF6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Integral group rings of solvable groups with trivial central units","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.GR","authors_text":"Andreas B\\\"achle","submitted_at":"2017-01-16T16:21:30Z","abstract_excerpt":"The integral group ring $\\mathbb{Z} G$ of a group $G$ has only trivial central units, if the only central units of $\\mathbb{Z} G$ are $\\pm z$ for $z$ in the center of $G$. We show that the order of a finite solvable group $G$ with this property, can only have $2$, $3$, $5$ and $7$ as prime divisors, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04347","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:11:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4wA5xZoHMPXT+WOCQgglSyzalUf6auIb7A0d7tGEe5qHCkrz8EGl0dHWHX8p99YHfreq1uk+94l5wq2VyWRMCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T12:19:46.481559Z"},"content_sha256":"97c77a03446d7de8c01789f73c42364ed21005bcf38ed55fe764e02e2336d6fd","schema_version":"1.0","event_id":"sha256:97c77a03446d7de8c01789f73c42364ed21005bcf38ed55fe764e02e2336d6fd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JGVDTCD5WQ6AKHYBXOTH436IF6/bundle.json","state_url":"https://pith.science/pith/JGVDTCD5WQ6AKHYBXOTH436IF6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JGVDTCD5WQ6AKHYBXOTH436IF6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T12:19:46Z","links":{"resolver":"https://pith.science/pith/JGVDTCD5WQ6AKHYBXOTH436IF6","bundle":"https://pith.science/pith/JGVDTCD5WQ6AKHYBXOTH436IF6/bundle.json","state":"https://pith.science/pith/JGVDTCD5WQ6AKHYBXOTH436IF6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JGVDTCD5WQ6AKHYBXOTH436IF6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:JGVDTCD5WQ6AKHYBXOTH436IF6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"040766acb811a7ce444f1b6d1e48198f8a488c5a33b77b46b288b00adbd06235","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-01-16T16:21:30Z","title_canon_sha256":"7729570c7d14161d1fb6699c5265ab15870458ccd56f7365bdc7920182c0042b"},"schema_version":"1.0","source":{"id":"1701.04347","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.04347","created_at":"2026-05-18T00:11:11Z"},{"alias_kind":"arxiv_version","alias_value":"1701.04347v4","created_at":"2026-05-18T00:11:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.04347","created_at":"2026-05-18T00:11:11Z"},{"alias_kind":"pith_short_12","alias_value":"JGVDTCD5WQ6A","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"JGVDTCD5WQ6AKHYB","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"JGVDTCD5","created_at":"2026-05-18T12:31:24Z"}],"graph_snapshots":[{"event_id":"sha256:97c77a03446d7de8c01789f73c42364ed21005bcf38ed55fe764e02e2336d6fd","target":"graph","created_at":"2026-05-18T00:11:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The integral group ring $\\mathbb{Z} G$ of a group $G$ has only trivial central units, if the only central units of $\\mathbb{Z} G$ are $\\pm z$ for $z$ in the center of $G$. We show that the order of a finite solvable group $G$ with this property, can only have $2$, $3$, $5$ and $7$ as prime divisors, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.","authors_text":"Andreas B\\\"achle","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-01-16T16:21:30Z","title":"Integral group rings of solvable groups with trivial central units"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.04347","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:22b5173426fc5c47b8a890c76e2f3094f2684b2f8b094f3629ae0967ef386eb1","target":"record","created_at":"2026-05-18T00:11:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"040766acb811a7ce444f1b6d1e48198f8a488c5a33b77b46b288b00adbd06235","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2017-01-16T16:21:30Z","title_canon_sha256":"7729570c7d14161d1fb6699c5265ab15870458ccd56f7365bdc7920182c0042b"},"schema_version":"1.0","source":{"id":"1701.04347","kind":"arxiv","version":4}},"canonical_sha256":"49aa39887db43c051f01bba67e6fc82fad561a20838c9436370850350bcf48e0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"49aa39887db43c051f01bba67e6fc82fad561a20838c9436370850350bcf48e0","first_computed_at":"2026-05-18T00:11:11.798596Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:11:11.798596Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"sSOPrOFfFA5eQ1yWMiwyIu7JFxwyoQbZUoK91H4GOf1xLgxT0BrgXlNVExhqFv+Z74NhwAPBP7GJlijoBgv5Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:11:11.799492Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.04347","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22b5173426fc5c47b8a890c76e2f3094f2684b2f8b094f3629ae0967ef386eb1","sha256:97c77a03446d7de8c01789f73c42364ed21005bcf38ed55fe764e02e2336d6fd"],"state_sha256":"1e27698784a46a148d2f9801c08fe081162c6058ec830394820bc90472118e21"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vJZJPow/KTIg/T0afMOOWBDhx2eqQ6ZI07YlB6XuSvvlE0sb/AUh9jdScTseY9mi/xbLxm2W+f9Ip346skWFDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T12:19:46.483413Z","bundle_sha256":"937a4069d711a74f62e59fb26b76ad269155d7d59ab9bfe200b7d06b5472f03d"}}